{"paper":{"title":"Hill's level surfaces in the circular restricted three-body problem solved","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Hill's level surfaces in the circular restricted three-body problem admit an exact closed-form expression obtained by inverting the Jacobi integral into a cubic equation.","cross_cats":[],"primary_cat":"astro-ph.IM","authors_text":"Jean-Marc Hur\\'e","submitted_at":"2026-04-23T08:42:22Z","abstract_excerpt":"We report the closed-form expression for Hill's surfaces in the circular restricted three-body problem. The solution $\\phi(r,\\theta)$, derived in the primary-centric spherical coordinate system, is deduced from a cubic equation delivering at most two roots on each side of a separatrix. The famous patterns (tadpole, horseshoe and peanut shapes, Roche lobes and Hill's quasi-spheres) are exactly produced."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We report the closed-form expression for Hill's surfaces in the circular restricted three-body problem. The solution φ(r,θ), derived in the primary-centric spherical coordinate system, is deduced from a cubic equation delivering at most two roots on each side of a separatrix.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the Jacobi integral level sets in the standard CR3BP effective potential can be inverted exactly into a cubic equation in primary-centric spherical coordinates without loss of information or hidden approximations.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A closed-form solution φ(r,θ) for Hill's surfaces in the CR3BP is obtained by solving a cubic equation that reproduces tadpole, horseshoe, Roche lobe, and quasi-spherical shapes.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Hill's level surfaces in the circular restricted three-body problem admit an exact closed-form expression obtained by inverting the Jacobi integral into a cubic equation.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7054b64ac10e64e3efcb53b8aaf88cff54f66e3bf5ce4a8871002209a470fd26"},"source":{"id":"2604.21426","kind":"arxiv","version":2},"verdict":{"id":"7fdd9a88-c46f-47ad-a351-0b3d266437aa","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T14:09:00.710844Z","strongest_claim":"We report the closed-form expression for Hill's surfaces in the circular restricted three-body problem. The solution φ(r,θ), derived in the primary-centric spherical coordinate system, is deduced from a cubic equation delivering at most two roots on each side of a separatrix.","one_line_summary":"A closed-form solution φ(r,θ) for Hill's surfaces in the CR3BP is obtained by solving a cubic equation that reproduces tadpole, horseshoe, Roche lobe, and quasi-spherical shapes.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the Jacobi integral level sets in the standard CR3BP effective potential can be inverted exactly into a cubic equation in primary-centric spherical coordinates without loss of information or hidden approximations.","pith_extraction_headline":"Hill's level surfaces in the circular restricted three-body problem admit an exact closed-form expression obtained by inverting the Jacobi integral into a cubic equation."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.21426/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":35,"sample":[{"doi":"","year":2026,"title":"Hill's level surfaces in the circular restricted three-body problem solved","work_id":"8e451d7d-3a61-499c-a807-92d2c8b892ce","ref_index":1,"cited_arxiv_id":"2604.21426","is_internal_anchor":true},{"doi":"","year":1982,"title":"V. Szebehely,Theory of orbits. The restricted problem of three bodies.(1982)","work_id":"","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1994,"title":"A. D. Bruno,The Restricted 3-Body Problem: Plane Pe- riodic Orbits(De Gruyter, Berlin, New York, 1994)","work_id":"","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2000,"title":"The restricted three- body problem,","work_id":"","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2007,"title":"J. M. Faidit,Limites et lobes de Roche(Vuibert, 2007)","work_id":"","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":10,"snapshot_sha256":"dea02f31ac7e3b7821e20331fd3e28f4f2bc41ba5a238f4464f39dd808371139","internal_anchors":1},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}